Yes — every US-registered aircraft has a specific engine manufacturer and model on file with the FAA, alongside the airframe's own make, model, and serial number. It is part of the same registration record you would pull for accident or ownership history, and it is worth checking, because it is one of the few places a records search can catch an engine that was swapped without the paperwork to match.
What the record actually shows
The filed engine record is the make and model of engine the aircraft is registered with — not a serial number, and not flight hours. It tells you what engine type the FAA has on file for that airframe, which lets you check one specific thing: does the engine you are looking at match the one that is supposed to be on it?
Why a mismatch matters
Engines get changed — for a legitimate upgrade, after a failure, or as part of a repair. Done correctly, an engine change is documented: an FAA Form 337 (Major Repair and Alteration) is filed, and if the new engine differs from what the type certificate allows, the change needs its own approval (a Supplemental Type Certificate, or STC). Done informally, none of that paperwork exists, and the airplane may be flying with an engine its own airworthiness certificate does not actually cover.
A mismatch between the registered engine and the one bolted to the airplane is not automatic proof of anything sinister — records lag reality sometimes, and paperwork gets missed on both legitimate and illegitimate changes. But it is a specific, answerable question to put to the seller before you buy: where is the 337, and where is the STC if one was required? If neither exists, that is worth knowing before the money moves, not after.
Where this fits in the order of checks
Like every other records check, this one is nearly free and takes minutes — pull it alongside registration status, accident history by serial number, and filed liens, before you spend money on travel or an inspection. It will not tell you the engine's condition or remaining time before overhaul; that is a logbook and inspection question. What it tells you is whether the airplane's paper trail agrees with the airplane in front of you, which is exactly the kind of question a $500–1,500 inspection is too expensive to be your first line of defense on.